**C. Nore****, W. Herreman, **L. Cappanera, J. Commenge, F. Luddens, J. Varela-Rodriguez, H. Zaidi, R. Zanella.

The magnetohydrodynamic (MHD) equations describing the motion of an electrically conducting fluid couple the velocity field and the magnetic induction by the Lorentz force and the Ohm's law. Using different codes developed at LIMSI coupled with innovative analytical models, we develop this subject along three main axes: flows in **liquid metal batteries** (Tayler instability, metal pad roll); the generation of a magnetic field by the turbulent motion of a conducting fluid (dynamo effect in the **von Kármán Sodium experiment**, optimization of flows to make them **dynamo-capable)**; thermal transfer by a ferrofluid in electrical transformers (thermo-magnetic convection in ferrofluids).

### Multiphase flows in liquid metal batteries

### Dynamo action in the Von Kármán Sodium experiment

The von Kármán Sodium experiment (or VKS http://perso.ens-lyon.fr/nicolas.plihon/VKS/index.php) investigates the magnetic field generation by a liquid sodium turbulent flow driven by two counter-rotating impellers consisting of disks and blades. It is the only one to have achieved dynamo regimes in 2010 showing time reversals of the magnetic induction like those of the earth's field, but for this it was necessary that the impellers be in soft iron. The role of the ferromagnetic material remains puzzling and we intend to get information on the underlying mechanism by relying on the SFEMaNS code we develop since 2002 (collab. J.-L. Guermond, TAMU, Texas). Scientific bottlenecks are first to take into account iron blades corresponding to an azimuthal variation of the magnetic permeability but also to reach the large kinetic Reynolds numbers of the flow (about 10 millions). One proposed method to remove the first bottleneck is to consider an average permeability axisymmetric and treat the azimuthal variations as a source term of the induction equation (PhD thesis of L. Cappanera). In the simplified case of kinematic dynamo where the velocity field is prescribed (chosen as the azimuthal and time-averaged turbulent experimental field), a numerical code developed at GeePs and based on the Whitney elements has led to progress in the understanding of the field of generation mechanism by ferromagnetic blades (postdoc of H. Zaidi funded by the Labex LASIPS ). Another key ingredient is the spiral vortex appearing behind each blade: it collimates any pre-existing magnetic field and amplifies it more when the blades and disks are made of ferromagnetic material than when they are made of conducting material (post-doc of J. Varela Rodriguez funded by an InterLabex contract).

To resolve the second bottleneck, a nonlinear stabilization technique was considered to achieve large kinetic Reynolds numbers. After the two approaches were validated, we have coupled them to compute the first three-dimensional numerical simulations of VKS at high kinetic Reynolds numbers.

*Velocity field (left) and magnetic field (right) for a DNS of the dynamo action in the von Kármán Sodium experiment Nore et al., Europhysics Letters, (2016) and Nore et al., Journal of Fluid Mechanics, volume 854, pp. 164--195 (2018)]. *

### Dynamo optimization

Given the difficulty of producing the dynamo effect experimentally, studying how to lower the magnetic Reynolds number threshold as much as possible is essential. Optimization studies have therefore always accompanied the experimental campaigns, but these optimizations always concern a small number of parameters. In a recent work (A. P. Willis, 2012, PRL), A. Willis applies variational optimization methods to find the most efficient flows in huge parameter spaces (+10^{5} parameters). This method has been adapted in LIMSI to study several new configurations. During her thesis (2013-2018) at ETH Zurich, L. Chen (co-supervised by W. Herreman) found the most effective dynamos in cases where the fluid is confined inside a cube (JFM 2015) and then in a sphere (JFM 2018). Figure below on the left shows a snapshot from these simulations in spherical geometry.

The same type of method is used to study the fragility of anti-dynamo theorems. For example, it is known that a pure shear flow can never be dynamo-capable and that, to trigger the dynamo, velocity disturbances of finite amplitude are required. Using the variational method, we have shown that a small velocity disturbance of amplitude 1/Rm added to the Kolmogorov flow is already sufficient to trigger a dynamo that reaches its threshold at Rm (JFM Rapids, 2016). In Figure below on the right, lines of the velocity field **u** of the minimum disturbance and the destabilized magnetic mode **B** are shown.

*(Left) Destabilized magnetic field generated by the optimal dynamo in a sphere [Chen et al., JFM (2018)]. (Right) Minimal perturbation u triggering a magnetic field (B) in the Kolmogorov flow [Herreman, JFM Rapids (2016)].*

### Thermo-magnetic convection in ferrofluid suspensions

We study the modelling and simulation of magnetic fluids in a thermal transfer context (PhD thesis by R. Zanella in co-supervision with GeePs laboratory, ED SMEMaG, funded by Labex LaSIPS). In particular, the thermo-magnetic convection cooling of transformers immersed in ferrofluids, consisting of a vegetable oil (electrically insulating, non-magnetic and environmentally friendly) and ferromagnetic nanoparticles, is being investigated. The mathematical model includes the following equations: magnetostatics, Navier-Stokes for a Newtonian and incompressible fluid and energy conservation. Due to magnetic nanoparticles, coupling terms appear: a forcing term in the Navier-Stokes equations (Kelvin force) and terms depending on the magnetic field in the energy equation (Joule heating and pyromagnetic term). The temperature variation of the physical properties of the ferrofluid has a strong influence. These developments have been implemented in the SFEMaNS code (collab. J.-L. Guermond, TAMU, Texas). The numerical study is based on different transformer models, ranging from a simplified model (a solenoid immersed in a tank filled with ferrofluid, see Figure below) to primary and secondary circuits of various geometries. A ferromagnetic and laminated core can be added to approximate a real transformer and increase the magnetic field in the fluid. The simplified model, experimentally studied at GeePs laboratory, allows to validate the numerical approach in the case of a vegetable oil. The same configuration with ferrofluid will be studied in the future.

*Scheme of the simplified model used in the GeePS experiment (left) and temperature iso-surfaces in degrees Celsius calculated with the SFEMaNS code (middle : vegetable oil ; right : ferrofluid). With ferrofluid, thermo-magnetic convection cells appear near the side walls and under the solenoid, reducing the temperature difference by increasing thermal exchanges [PhD thesis by R. Zanella in co-supervision with GeePs, ED SMEMaG, funded by Labex LaSIPS].*